The homotopy infinite symmetric product represents stable homotopy

Abstract

We modify the definition of the infinite symmetric product of a based space $X$ by applying the homotopy colimit instead of the colimit. This gives a topological monoid $S{P}_h\left(X\right)$ and using formal properties of homotopy colimits, we prove that its group completion represents the stable homotopy of $X$. In this way we get a streamlined approach to the Barratt–Priddy–Quillen theorem.